B747-400 Captain reported engine malfunction caused them to shut down the engine and divert to nearest appropriate airport.

Narrative

I was the PF on this leg to ZZZ. FO Person A was the PM. FO Person B was in the third seat with FO Person C behind me. At thrust reduction after takeoff (NADP (Noise Abatement Departure Procedure)) the aircraft yawed to the left. I countered with right rudder and trimmed the aircraft. I then noticed that engine #1 had reduced thrust (85% N1) with an over temp. I reduced thrust on engine #1 to lower the EGT (Exhaust Gas Temperature). This completed the Quick Reference Checklist items. The engine was still producing thrust with no indication of imminent failure so; I engaged the autopilot and elected to continue the departure and climb with the concurrence of my crew. During the climb we devised a plan to continue our flight; exit foreign airspace; and divert to ZZZZ. Additionally; we continued to monitor engine #1 for any signs of imminent failure. Once above 10000' I called Dispatch (Dispatcher; MX (Maintenance) control; and duty pilot were all on the line) and advised the situation and our plan to exit Foreign airspace with the engine still operating and producing thrust and then divert to ZZZZ once we exited.They all agreed with our plan. We climbed to 8100 meters for the remainder of our time in Foreign airspace. Also; we completed the Abnormal Engine Indication checklist up until shutting down the engine. Since engine #1 was still operating we climbed above the altitude by about 1000'. Upon checking in with ZZZ [ARTCC] control we advised with an engine problem and that we'd like to hold and dump fuel followed by a diversion to ZZZZ. We were given a reroute to ZZZZ followed by instructions to hold south of the ZZZ VOR at 20 nm. On our way to the hold we descended from FL270 to FL250 which put us below altitude. During this time we discussed landing overweight and we all agreed that the conditions were in our favor to do so. The weather was VFR; we were rested; had no other failures; and we had plenty of time to develop our plan. We ran several performance calculations and settled upon landing at 330000 kg. After entering the hold we completed the Abnormal Engine Limit checklist and shut down the #1 engine. Following that we determined how much fuel to dump to arrive at our planned weight of 330000kg. We completed the fuel dump checklist and advised ATC when we began dumping fuel. During the dumping procedure we agreed we should land on Runway XXR for the shortest taxi route to parking considering what our brake temps might be. Also; I briefed the arrival and approach using our standard procedures. Once the fuel dump was completed we finished our checklists and advised ATC this we were ready to begin the arrival. We also had another call with Dispatch to update our ETA to ZZZZ. The arrival and approach were normal and uneventful.I made a smooth landing on Runway XXR with flaps 25 and autobrakes 3. I also used full reverse on engines 2 and 3 during rollout. Following the landing we taxied to parking and advised the ground crew of possible hot brakes on all wheels. Notes/Lessons Learned: We had time on our side. This contributed to a positive outcome. My crew was resourceful and a fantastic group to work with. We had nearly zero stress during this event. I decided as Captain to set certain 'checklist' during this flight. We would accomplish certain items before each checklist and then make a decision about what to accomplish next and if we were safe to proceed. As an example we had 3 distinct portions of this flight which ended with a checklist to the next portion. 1. Takeoff to entering foreign airspace. 2. Entering foreign airspace to beginning the arrival. 3. Beginning the arrival to a safe landing. By doing this we were able to manage the situation effectively and not become overwhelmed with stuff to do. Everyone remained calm; provided valuable input; and we worked as a team. It was really quite boring to be honest; as it should be. Dispatch; MX control; and the Duty Pilot were helpful and provided support as needed and expected. Many thanks to the team on the ground. We could have landed at a much higher weight than we did thus saving more fuel. The brakes never got above 3 and stopping after landing was never in doubt. Maybe 340-345 tons would have been the limit considering landing distance and brake temps. On the other hand; it was great to see the actual performance of the aircraft in this situation. It was a real confidence builder for everyone. FO Person C in the fourth seat had difficulty communicating with the rest of the crew because there are no MIC jacks in that seat. He was literally writing notes and passing them to FO Person B while this was going on. While FO Person C contributed greatly to the outcome the inability to communicate thru the intercom really hindered his ability to help. We need to explore how we can fix this issue on those aircraft without the MIC jacks installed.Cause:Suspected failure of THE or PART of the turbine section of engine number 1. Post flight inspection revealed debris in the turbine exhaust of engine number 1.Suggestions:FO Person C in the fourth seat had difficulty communicating with the rest of the crew because there are no MIC jacks in that seat. We was writing notes and passing them to FO Person B while this was going on. While FO Person C contributed greatly to the outcome the inability to communicate thru the intercom hindered his ability to help. We need to explore how we can fix this issue on those aircraft without the MIC jacks installed in the fourth observer position.